$($a$)$ State the Cauchy$-$Schwarz Inequality.

$($b$)$ Treat the quotes $Q_1,Q_2,Q_3,Q_4$ below as four separate documents.

$Q_1$ : The greatest glory in living lies not in never falling, but in rising every

time we fall. Nelson Mandela

$Q_2$ : Many of life's failures are people who did not realize how close they were

to success when they gave up$.$ Thomas A$.$ Edison

$Q_3$ : The secret of life is to fall seven times and to get up eight times. Paulo

Coelho

$Q_4$ : It is impossible to live without failing at something unless you live so

cautiously that you might as well not have lived at all, in which case you have

failed by default. J$.$K$.$ Rowling

Treat all words with the stem fail fall, failure, failing, failed $\{\begin{array}{c}?\end{array}$ as

equivalent, and of all words connected to the word life life, living, live,

lived $\})$ as equivalent.

For the four documents $Q_1,Q_2,Q_3,Q_4,$ and the two families of terms $T_1$

and $T_2,$ write down the $24$ term$-$by$-$document matrix and the normalised

term$-$by$-$document matrix.